Method and apparatus for computation



July 31, 1928.

W. KOENIG. JR

METHOD AND APPARATUS FOR GOHPUTATION Filed. Dec 3Q, 1925 4 Sheets-$heet1 A TTORNEY July 31, 1928. 1,678,674 W. KOENEG, JR

METHOD AND APPARATUS FOR-q-COIPUTATION Filed 090450, 1925 4 Shets-Sheet5 CORRECTIONS n SUM n READING READING SUM a a FLAmHETER READINGPLANIMETER rmmns INVENTOR W Jr.

BY r

A TTORNE Y July 31, 1928. 1,678,674 W. KOENIG, JR

METHOD AND APPARATUSJ OR COMPUTATION Filed Dec. 50, 1925 4 Sheets-Sheet4 IN VEN TOR L Y m ZRNE Y Patented July 31, 1928. k

UNITED STATES, 'PAT-ENTNOFFICE.

WALTER KOENTG, .TR., OF NEW-YOR K, N. Y., ASSIGNOB TO AMERIOAN TELEPHONEAND TELEGRAPH; COMPANY, A CORPORATION OF NEW YORK.

METHOD AND APPARATUS FOR COMPUTATION.

' Application filed December 30, 1925. Serial No. 78,470.

An object of m invention is to provide new and improve apparatus and acone spondin method to facilitate the computation of t e amplitudes ofvarious harmonics 6 in a composite periodic curve. Another object of myinvention is to provide apparatus and method for computing thecoeflicients of a Fourier series representingthe variation of a givenmagnitude with time. Another 10 object of m invention is to provide forutilizing a p animeter for rapid summation of lengths on a diagram in acertain way. These objects and-various other objects of m invention willbecome apparent on consideration of an example of practice according tothe invention. In the following specification I will disclose certainapparatus and a certain method in illustration of my invention. It willbe understood that the following specification will relate to thisparticular example of the invention and that the invention will bedefined in the appended claims.

Referring to the drawings, Figure 1 is a top plan view of apparatus thatmay be em ployed in the practice of my invention; Fig. 2 is an endelevationlooking in the direction of the arrow 2 of Fig. 1; Fig. 3 is adetail sectional elevation taken on the line 3 of Fig. 1; Fig. 4. isanother detail sectional elevation taken on the line 4 of Fig. 1; Fig. 5is a diagram showing a comparatively simple curve and illustrating astep of my method in relation thereto; Fig. 6 is a scale in differentpositions to be considered relatively to Fig. 5; Fig. 7 is a form inwhich the steps of computation may be entered for the analysis of acurve such as that shown in Fig. 1; Fig. 8 is a diagram showing theamplitudes of various harmonics of said curve of Fig. 1 as determinedbythe practice of my invention; and Figs. 9 and 10 are special diagrams toillustrate certain steps in the performance of the method as disclosed.

Referring to Fi s. 1 to 4, the drawing board 1 has guides? and 7 fixedon its ends by bracketssuch as 8.] At their front ends these guides haveholes forming bearings for the shaft 3, carrying the sprocket wheels 5and 5' at its ends. Adjustable supports 6 and 6 on the ends of the board1 also carry bearings for the parallel shaft 2 with the sprocket wheels4 and 1 at its ends. The chains 10 and 10 connect the sprocket wheels 4,5 and 4, 5', respectively. On one end of the front shaft 3 is ahandwheel by which the shaft may be rotated, thus driving the chains 10and 10'.

The shoe 12'1314 slides on the guide 7, and there is a similar shoe onthe guide 7 These shoes are rigidly connected with the ends of across-bar 15. Each shoe has a lug 9 (or 9) engaging a link of the chain10.

Each shoe member 13 carries a clamping 5 screw with a handle 17 and ashoulder 18 beneath which is clamped the notched scale 16.

The curve to be analyzed is photographicallyenlarged so that its eriodlength will be definite, say one foot. uch a curve 39 is shown on thesheet 20, fastened by thumb tacks 21 on the board 1.

The planimeter shown in Fig. 1 is of fa- H miliar design comprising theointed arms 22-and 24, and with its fixed pivot under the weight 23. Theplanimeter wheel adapted to slide and rotate is shown at 25 on the shaft26 with the scale 27 reading against the Vernier 28; 29 is a worm wheelengaging a worm on the shaft 26 and adapted to count complete rotationsof the wheel 25. The planimeter tracing stylus 19 is directly under thehandle 30.

The cross-bar 15 carries a slider 31, which may be moved by means of thehandle 32. Projecting from the slider is a bar 33 notched at its end 34,and the planimeter handle 30 lies in this notch and is held snuglyby thespring 35. 36 is a spring on the bar 33, lying under the scale 16, andadapted to support 1t if it should tend to sag. Projecting from theslider 31 is a leaf spring 37 carrying a pin 38 adapted to engage anotch of the-scale 16. The notches in the scale 16- are shallow andbevelled so that while the I engagement of the pin 38 with one of themtends to hold the slider 31 at the corresponding position on the bar 15,nevertheless a push on the handle 32 along the bar 15 will disengage thepin 38 from a notch.

40 and 41 are two points'of the curve 39.

curve. I will illustrate the use of the fore going described apparatusby explaining how the coefficients for the fifth harmonic of this curvemay be obtained. When I refer to a harmonic by number, 1 count thefundamental as the first harmonic, for example, what 1 call the fifthharmonic is the fourthabove the fundamental.

Fouriers series may be written in the fa- 20 miliar forms 1 +B2 s n2r+-----+B.. sin m+---- 1 and 25 /A,,+B sin m+tan' (2) and we seek thecoeflicients A and B in equation (1) for Fig. 5.

AF is any line parallel to the m-axls of the curve corresponding to theline 1O4;1 of Fig. 1. There are various scales 1G for the variousharmonics. The scale for the fifth harmonic as shown at 51 in Fig. 6 isplaced in the machine as indicated at 16 in Fig. 1. The planimetertracing stylus 19 is set at A, and the scale is moved lengthwise tillits zero notch engages the pin 38. Then the scale is clamped at its endsby means of the screws 17 The planimeter is set at zero and thehandwheel 11 is rotated, thus moving the planimeter stylus 19 up till itmeets the curve at A. Then the handwheel 11 is held fixed and the slider31 is moved until its catch 38 engages the next notch 1 (in Fig. 5).Thus the planimeter tracing stylus 19 is carried along the line A BinFig. 5. Next the handwheel 11 is turned until the planimeter stylus 1,9meets the curve as at B, the stylus 19 moving along the path B B in Fig.5. Then by a movement of the slider to notch 2 of the scale, the tracingstylus 19 goes along the path B C. Similarly, the movement is continuedalong the full line path C C D D E E F F, the point F being on the lineA F parallel to the curve-axis.

Then the planimeter tracing stylus 19 is moved along the line A F from Fto G as determined by notch 4 half way between notches 4 andb. Then theslider 31 is held and the scale is loosened and displaced to the lefttill notch 5 reengages the pin 38 on the slider, just as they previouslyengaged at position G. The new position of the scale is indicated 3* 52in Fig. 6. Having clamped the scale in this new position, the tracingstylus 19 is moved up from G to G and the dotted line path is tracedback in the manner that will be well understood from what has been said,G G H H K K L L A.

If it should be known that harmonics of order higher than the fourteenthare absent or negligible, then the final reading of the planimeter at Adivided by twice the length of the base line AF will be the value of theeocliicient A If harmonics higher than the fourteenth are present andnot negligible, then the reading of the planin'ieter will be subject toa correction that will be e.\'- .plained presently.

To get the coei'licient B the stylus 19 is held at A and the scale isagain loosened and shifted to the right of its original position aquarter of the distance between the usual scale divisions, that is, tothe position shown at 53 in Fig. 6,-and a new start is made from A,going first to the initial notch now at A ,and thereafter the procedureis as described above from this point until the planimeter tracer isbrought back to A, and (subject to the same possible need" forcorrection) the planimeter reading divided by twice the length of thebase line AF will be the value Of Specifically to illustrate thepractice of my invention, I show in Fig. 7 a form used in the analysisof oscillograph records of voice curves. In these cases the harmonicsare of considerable magnitude and importance, and Fig. 7 is designed todeal with all the harmonics as high as the thirtieth. The procedure isto get the thirtieth harmonic first and the lower harmonics successivelyin decreasing order. Hence the scale 16 for the thirtieth harmonic isused first. The procedure is similar to that already described for thefifth harmonic. It is assumed that harmonics higher than the thirtiethare negligible. Therefore the value +0.07, which is the final planimeterreading at A, is entered directly in Fig. 7. Represent this planimeterreading by a it must be divided b twice the length Z of the base 4041(in ig. 1)- to give A In general lln= AnX2l (3) and similarl bu=Bn X21With Z equal to 12 inches and the planimeter reading in inches Since weare concerned with the relative amplitudes of the harmonics, the directplanimeter readings a are entered in the tableFigure 7, and the value a+0.07 will be found in the column headed a and onthe line marked 30 atthe left. Similarly for the value Z2 +0.02.

Having determined the thirtieth harmonic,

the scale 16 therefor is replaced by the scale for the twenty-ninthharmonic the values a ,,=+0.34 and b ,,=.+0.14 are obtained and enteredas shown in Fig. 7. ThlS procedure is repeated for the lower harmonicsin descending order and eventually the values 1.85 and 2.65 are obtainedand entered for (1 and b respectively.

For harmonics from the thirtieth to the sixteenth inclusive, I employ 15respective scales. For the fifteenth harmonic, I use everyother notch ofthe scale for the thirtieth, for the fourteenth every other notch of thetwenty-eighth, and so on.

Proceeding in the same way using the proper scale for the tenthharmonic, the planimeter readings are -7.30 and 0.98 but I designatethese a and 6 because they are subject to correction as compared with aand b In general, any planimeter readings a and b are 'subject'tocorrection according to the equations n= n ln ln 7 (6) b.=b.'+b|.b-. n-1 It will be seen that each planimeter reading is to be corrected byhigher harmonic amplitudes, the lowest of which has an order three timesthe one for which the planimeter readings are taken. Accordingly thelowest order correction for the thirtieth harmonic is the ninetieth, forthe twenty-ninth it is the eighty-seventh and so on, and for theeleventh it is the thirty-third. Having assumed that all harmonicshigher than the thirtieth are negligible, no corrections have been madefor the planimeter readings hitherto. In other words, Equations (6) and(7) reduce to a =a and b =b nearly enough for practical purposes forn=11, 12.....29, 30. But for n=10, the Equations (6) and (f7) become a'=a a and b =b +,b or practical purposes. Accordingly, the planimeterreadings 7 .30 and-0.89 obtained for a and 6 respectively, are enteredon line 10 under the respective column headings a and b. The

roper corrections are copied in the adoining spaces and the correctedvalues are entered in the columns headed a and b.

Proceeding similarly, the columns a and b and (1 and b are filled on upthrough for the seventh harmonic." For the practical corrections for thesixth harmonic Equations .(6) and (7) reduce to nearly enough forpractical purposes. The resultant corrections +0.23 and 0.27 arecomputed as indicated in the upper right cornor of Fig. 7 and carriedover to the indica ted places adjacent the planimeter readings withwhich they are combined to give the corrected values -3.27 and +0.08 inthe columns a and 12. Similarly, the columns a, b, a and b of the tableof Fig. 7 are filled on up through the values for the third harmonlc,using the corrections computed as in dicated at the right of Fig. 7. Forthe third harmonic Iuse the scale for the twentyfourth and move theslider -8 notches at a time.

For the second and first harmonics, a similar procedure could be used,but this would place the point L of Fig. 6 at an inconvenient distance.Accordingly I follow a special procedure, using intermediate notches onthe scale for the sixteenth harmonic as shown in Figs. 9 and 10, whichwill be readily understood from the following equations on Fig. 9.Planimeter readings M M O O M M+Q, Q/S S Q, a planimeter readings N N PP N" N +R R T T R R= b and on Fig. 10, planimeter reading M, M Q""'{ Q,M M= g 9 plgn imeter reading 0 O S S 0" various harmonics. One suchnumber, 3.23,

is shown, for example, in Fig. 7 for the eleventh harmonic. All theseresultant numbers are plotted as ordinates in Fig. 8. The absolute termA if desired, may be obtained by measuring y (Fig. 5) in inches,doubling to get 2y. subtracting a,, and adding the correction for a(indicated by the encircled 6 at the bottom of Fig. 7). The result, a=24A.,. The method and apparatus herein disclosed will serve to obtainthe harmonics of a composite periodic function with much less labor thanby direct numerical computation. By my invention a practiced operatorcan get all the harmonics to the thirtieth in about three hours whereasby numerical comutation several days would be required.

e results by my method are accurate to 1 130 percent or better,depending on the accuracy 1nd sharpness of the enlarged curve to beanalyzed. I y '1 I claim: 1. A harmonic analyze-rKcomp-rising a boardadapted to receive an impression of the curve to be analyzed, aplanirneter on the board, a bar across the board parallel with the curveaxis, means to move the bar transversely of its length, and a slider onthe bar engaging the planimeter tracing stylus.

2. A harmonic analyzer comprising a board adapted to receive animpression of the curve to be analyzed, a ,plammeter on the board, a baracross. the board parallelwith the curve axis, means to move the bartransversely of 1ts length, and interchangesprocket wheels on theirends, two chains on said sprocket wheels, and a slider on the barengaging the planimeter tracing stylus. In testimony whereof, I havesigned my name to this specification this 29th day of December, 1925.

WVALTER KOENIG, JR.

